Left Regular Representation of Gyrogroups

Teerapong Suksumran
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Teerapong Suksumran: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Mathematics, 2019, vol. 8, issue 1, 1-9

Abstract: In this article, we examine a subspace L gyr ( G ) of the complex vector space, L ( G ) = { f : f is a function from G to C } , where G is a nonassociative group-like structure called a gyrogroup. The space L gyr ( G ) arises as a representation space for G associated with the left regular representation, consisting of complex-valued functions invariant under certain permutations of G . In the case when G is finite, we prove that dim ( L gyr ( G ) ) = 1 | γ ( G ) | ∑ ρ ∈ γ ( G ) | Fix ( ρ ) | , where γ ( G ) is the subgroup of Sym ( G ) generated by a class of permutations of G and Fix ( ρ ) = { a ∈ G : ρ ( a ) = a } .

Keywords: linear representation of gyrogroup; gyrogroup action; gyrogroup character (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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